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Extension of Khan’s homotopy transformation method via optimal parameter for differential difference equations. (English) Zbl 1442.65320

Summary: A new scheme, deduced from Khan’s homotopy perturbation transform method (HPTM) [Y. Khan, “An algorithm for solving nonlinear differential-difference models”, Comput. Math. Model. 25, No. 1, 115–123 (2014; doi:10.1007/s10598-013-9212-z); Y. Khan and Q. Wu, Comput. Math. Appl. 61, No. 8, 1963–1967 (2011; Zbl 1219.65119)] via optimal parameter, is presented for solving nonlinear differential difference equations. Simple but typical examples are given to illustrate the validity and great potential of Khan’s homotopy perturbation transform method (HPTM) via optimal parameter in solving nonlinear differential difference equation. The numerical solutions show that the proposed method is very efficient and computationally attractive. It provides more realistic series solutions that converge very rapidly for nonlinear real physical problems. The results reveal that the method is very effective and simple. This method gives more reliable results as compared to other existing methods available in the literature. The numerical results demonstrate the validity and applicability of the method.

MSC:

65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems

Citations:

Zbl 1219.65119
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[1] Fermi, E.; Pasta, J.; Ulam, S., Collected Papers of Enrico Fermi II (1965), Chicago, Ill, USA: University of Chicago Press, Chicago, Ill, USA
[2] Gepreel, K. A.; Nofal, T. A.; Alotaibi, F. M., Exact solutions for nonlinear differential difference equations in mathematical physics, Abstract and Applied Analysis, 2013 (2013) · Zbl 1282.39011
[3] Gepreel, K. A.; Nofal, T. A.; Al-Thobaiti, A. A., The modified rational Jacobi elliptic functions method for nonlinear differential difference equations, Journal of Applied Mathematics, 2012 (2012) · Zbl 1269.34004
[4] He, J.-H., A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International Journal of Non-Linear Mechanics, 35, 1, 37-43 (2000) · Zbl 1068.74618
[5] He, J.-H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26, 3, 695-700 (2005) · Zbl 1072.35502
[6] Ganji, D. D.; Sadighi, A., Application of He’s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 4, 411-418 (2006)
[7] He, J.-H., Homotopy perturbation method for solving boundary value problems, Physics Letters A, 350, 1-2, 87-88 (2006) · Zbl 1195.65207
[8] Mohamed, M. S.; AL-Malki, F.; Altalhi, N., Analytic and approximate solutions of time and space fractional nonlinear cubic equations via Laplace transform, Jokull Journal, 64, 6, 490-503 (2014)
[9] Gepreel, K. A.; Mohamed, M. S., An optimal homotopy analysis method nonlinear fractional differential equation, Journal of Advanced Research in Dynamical and Control Systems, 6, 1, 1-10 (2014)
[10] Wazwaz, A.-M., The variational iteration method: a powerful scheme for handling linear and nonlinear diffusion equations, Computers & Mathematics with Applications, 54, 7-8, 933-939 (2007) · Zbl 1141.65077
[11] Khan, Y.; Austin, F., Application of the Laplace decomposition method to nonlinear homogeneous and non-homogeneous advection equations, Zeitschrift für Naturforschung A, 65, 1-5 (2010)
[12] Madani, M.; Fathizadeh, M.; Khan, Y.; Yildirim, A., On the coupling of the homotopy perturbation method and Laplace transformation, Mathematical and Computer Modelling, 53, 9-10, 1937-1945 (2011) · Zbl 1219.65121
[13] Khan, Y.; Wu, Q., Homotopy perturbation transform method for nonlinear equations using He’s polynomials, Computers & Mathematics with Applications, 61, 8, 1963-1967 (2011) · Zbl 1219.65119
[14] Abidi, F.; Omrani, K., The homotopy analysis method for solving the Fornberg-Whitham equation and comparison with Adomian’s decomposition method, Computers & Mathematics with Applications, 59, 8, 2743-2750 (2010) · Zbl 1193.65179
[15] Khader, M. M., Numerical solution for discontinued problems arising in nanotechnology using HAM, Journal of Nanotechnology & Advanced Materials, 1, 1, 59-67 (2013)
[16] Yıldırım, A., He’s homotopy perturbation method for nonlinear differential-difference equations, International Journal of Computer Mathematics, 87, 5, 992-996 (2010) · Zbl 1192.65102
[17] Jagdev, S.; Devendra, K.; Sunil, K., A reliable algorithm for solving discontinued problems arising in nanotechnology, Scientia Iranica, 20, 3, 1059-1062 (2013)
[18] Obidat, Z.; Momani, S., The homotopy analysis method for solving discontinued problems arising in nanotechnology, World Academy of Science, Engineering and Technology, 52, 891-894 (2011)
[19] Hosseinzadeh, H.; Jafari, H.; Roohani, M., Application of Laplace decomposition method for solving Klein-Gordon equation, World Applied Sciences, 8, 7, 809-813 (2010)
[20] Faraza, N.; Khana, Y.; Austinb, F., An alternative approach to differential-difference equations using the variational iteration method, Zeitschriftfuer Naturforschung, 65a, 1055-1059 (2010)
[21] Khan, Y.; Vzquez-Leal, H.; Faraz, N., An auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations, Applied Mathematical Modelling, 37, 5, 2702-2708 (2013) · Zbl 1352.65172
[22] Khan, Y.; Wu, Q., Homotopy perturbation transform method for nonlinear equations using He’s polynomials, Computers & Mathematics with Applications, 61, 8, 1963-1967 (2011) · Zbl 1219.65119
[23] Khan, Y.; Madani, M.; Yildirim, A.; Abdou, M. A.; Faraz, N., A new approach to Van der Pol’s oscillator problem, Zeitschrift für Naturforschung A, 66, 620-624 (2011)
[24] Khan, Y.; Vazquez-Leal, H.; Hernandez-Martínez, L., Removal of noise oscillation term appearing in the nonlinear equation solution, Journal of Applied Mathematics, 2012 (2012) · Zbl 1251.34059
[25] Gepreel, K. A.; Mohamed, M. S., Improved (G′/G) expansion method for solving nonlinear difference differential equations
[26] Khan, Y., An algorithm for solving nonlinear differential-difference models, Computational Mathematics and Modeling, 25, 1, 115-123 (2014)
[27] Gepreel, K. A., Jacobi elliptic numerical solutions for the time fractional Boussinesq equations, Journal of Partial Differential Equations, 27, 3, 189-199 (2014)
[28] Gepreel, K. A., Exact solutions for nonlinear Toda Lattice difference differential equation, World Applied Sciences Journal, 27, 1792-1797 (2013)
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